Development and validation of two learning outcome measures: ‘Disposition to enter HE’ and
‘disposition to study mathematically demanding subjects in HE’
Julian Williams, Maria Pampaka, Laura Black, Pauline Davis, Paul Hernandez-Martinez, Geoff
Wake and Julian Williams
School of Education,
University of Manchester
Abstract
The aim of this paper is to describe and validate the development of two measures constructed to
measure AS students disposition (i) to enter HE and (ii) to further study mathematically-demanding
subjects, which we regard as potentially significant variables in monitoring or even explaining
students progress in to different studies in HE. The items for the scale were constructed on the basis
of interview data, and drew on a model of disposition as socially- as well as self- attributed.
Drawing on Rasch analyses of pilot and ‘main’ data sets, we find that the two scales each produce
healthy one-dimensional fits on what we take to be a ‘strength of commitment to enter HE’ and
‘disposition to study mathematically-demanding subjects further’ respectively. However, as a
measurement scale for this sample in this context the former scale suffers from a ceiling effect: our
sample are overwhelmingly committed to entering HE (at the early stages of AS level mathematics
course anyway). To ‘correct’ for this, we added some harder items to the analysis at a later data
point, and found (i) an item that improved separability of the instrument for the higher scorers, and
(ii) a massively misfitting, hard item that is worthy of future research.
1.
Introduction
This paper is based on quantitative analyses of a data-set from the ESRC TLRP research project on
widening participation in HE, ‘Keeping open the door to mathematically-demanding F&HE
programmes’. We particularly draw on sets of responses to a questionnaire administered to students
undertaking AS Maths and AS Use of Maths courses, and focus on the development and validation
of two new instruments for measures of students disposition towards HE and further studying
mathematically demanding subjects.
The economic significance of mathematics and the shortage of mathematically well-qualified
students and graduates (i.e. the ‘Mathematics Problem’) is strongly emphasised by recent reports
(i.e. Smith, 2004). Hence, we need to understand how mathematics can become more accessible to
students, especially those students for whom AS/A2 mathematics is a barrier to progressing into
mathematically demanding courses that confer social, cultural and economic capital. The general
aim of the research project is to understand how to widen participation in mathematically
demanding subjects generally, but particularly for our ‘target’ students, i.e. those students who are
at the margins of continuing with maths.
Towards this end we encountered the need for measures of students’ ‘perception of intention to
study in HE’, and additionally measures of their intention to persist in study of mathematicallyrelated topics in Further and Higher Education. To our knowledge there is as yet no existing
measure of intention to persist in the study of mathematically related topics in F&HE. Nor is there a
general measure of intention to enter HE that has been validated in F&HE that meets our
requirements. However, there is an eclectic but relevant literature that informs the development and
1
validation of such educationally socio-culturally sensitive measures (i.e. Eley and Meyer, 2004;
Hoyles, et al, 2001).
The first instrument, namely ‘disposition to enter HE’ consists of four statements eliciting students
own expectation about themselves going to university and the expectations of others about this
possibility (family, friends, teachers). Our view, supported by interviews, was that students might
be said to exhibit a stronger commitment to HE if they said that significant others had such
expectations for them (though in fact we noticed that they were sometimes less sure of their
teachers’ views than they were of their family and friends!)
The second instrument consists of 6 items aiming to capture information on students’ dispositions
to studying mathematically demanding subjects in future HE. The items included in both
instruments are presented to students in a multiple choice format and have various numbers of
response categories. This had direct implications for the selection of the appropriate (partial credit)
measurement model to be selected when calibrating these instruments. Validation was performed
by employing the Rasch Partial Credit Model (Bond & Fox, 2001) on a pilot sample of the project
(N=314) and suggested robust measures. Some problems appeared regarding the HE disposition
instrument, because of sample characteristics, i.e. high tendency of the particular group to report a
disposition of “going to HE”, and we tried to overcome these with additional items in the main
study sample. We will report on both these results in this paper.
We plan to use these validated soft measures as explanatory variables for exploring the
effectiveness of different FE maths programmes, as well as a predictor of students’ future decisions
/ choices at UCAS, in the next paper (Pampaka et al., this symposium). We finally discuss the
possibility that these instruments will have utility in the wider widening participation research
community.
2.
The Rasch Partial Credit Model
George Rasch, realized that, to be of any use at all in a measurement model, a measure must retain
its quantitative status, within reason, regardless of the context in which it occurs. Like a yardstick,
each test or any other construct’s item must maintain its level of difficulty, regardless of who is
responding to it. It also follows that the person measured must retain the same level of competence
or ability regardless of which particular test items are encountered, so long as whatever items are
used belong to the calibrated set of items which define the variable under study. Rasch also
recognized that the outcome of an interaction between an object-to-be-measured, such as a person,
and a measuring-agent, such as a test item, cannot, in practice, be fully predetermined but must
involve an additional, unavoidably unpredictable, component (Wright & Linacre, 1989).
The Rasch model (named after its developer) provides the means for constructing interval measures
from raw data. When data can be selected and organized to fit a Rasch Model, the cancellation
axiom of additive conjoint measurement is satisfied, a perfect Guttman order of response
probabilities and hence of item and person parameters is established, and items are calibrated and
persons measured on a common interval scale. The model proposes a mathematical relationship
between a person’s ability, the difficulty of the task, and the probability of the person succeeding on
that task (Wright & Mok, 2000; Acton, 2003; Wright, 1999).
The family of Rasch models is the only one that solves measurement problems, because these
models produce linear measures, overcome missing data, give estimates of precision, have devices
for detecting misfit and the parameters of the object being measured and of the measurement
instrument are separable (Wright and Mok, 2000). As Masters also (undated) notes a unique feature
2
of the Rasch model is that “when data fit the Rasch Model, it is possible to compare abilities
without knowing, or even having to estimate, the difficulties of the task” (p. 23).
Applications of Rasch Measurement, among others, include Item banking, test design, tailored
testing, self-tailoring, response validation (through the analysis of fit) and item bias (Wright, 1999)
which can be realized with different statistical analyses. When the name ‘Rasch Model’ appears in
connection with statistical analyses, it often means the so-called ‘dichotomous Rasch Model’ or the
‘one-parameter logistic model’ which is the simplest format of the models and records only two
levels of performance on an item, e.g. ‘Fail/Pass’.
The Partial Credit Model (PCM), which is used for the purposes of this paper, is considered to be an
extension of the simple dichotomous (Rasch) model, for outcomes recorded in more than two
ordered response categories, i.e. by awarding ‘partial credit’ for responses that are neither correct
nor totally incorrect. The model can be applied to any set of test or questionnaire data collected for
the purposes of measuring abilities, achievements, or attitudes provided that responses to each test
or questionnaire item are scored in two or more ordered categories (Masters, 1982,1999). Bode
(2001) lists three specific situations in which the PCM can be used, and those are when instruments
contain items:
with varying degrees of correctness for responses that can be ordered from least correct to most
correct,
that can be broken into component tasks, the first of which must be completed before the next is
attempted, and each of which can be scored as correct or incorrect,
where increments in the quality of performance are rated.
Athanasou and Lamprianou (2002) clarify what is additionally needed (compared to the simple
model) in order to draw the graphs for PCM (with the score probability lines for a question/item):
whereas one parameter for the people (the ability) and one parameter for the items (the difficulty of
getting 1 instead of 0) is enough for the simple model, in the partial credit model additional
parameters should be introduced as the δ (difficulty) parameter cannot describe the question fully. It
is essential to know the difficulty of achieving each of the score categories.
The PCM specifies that each item has its own rating scale structure, that is, the transition from one
category to the next can have a different meaning from one item to the next (Wright, 1999; Bode,
2001). In the following sections we will show how this model was employed to validate the two
measures of disposition.
3.
Results
The subjects (sample description and data collection)
Results presented in this paper are based on analysis of data from (i) a pilot study, and (ii) the first
two data points of the longitudinal, main stages of the project (DP1 and DP2) as shown below in
BOLD under ‘dispositions’. Note that these two instruments are called HEdisp and MathDis: the
other disposition instrument is a Maths self-efficacy instrument that we have reported elsewhere.
Table 1: Design of data collection
Background
Variables
Hard LOs
Family-in-HE
Year 1 (2006-7)
DP 1
DP2
Year 2 (2007-8)
DP3
Start of AS
End of AS
Start of A2
GCSE
grades
AS grades
3
LPN by
postcode,
EMA,
Gender
Language(Eng/
non-Eng/bi),
Ethnicity,
College
Dispositions:
HE Dis
HEDis-1
HEDis -2
HEDis -3
MathDis
MathDis-1
MathDis-2
MSE
Intentions/
Choices/
Decisions
MSE-1
Uni-int 1
STEMint1
MSE-2
Uni-int 2
STEMint2
MathsDis3
MSE-3
Uni-int 3
STEMint3
Pilot Data came from 313 AS student from 23 different further education institutions in UK (their
distribution regarding gender and course is shown in Table 2). 27 GCSE students (i.e. students who
have not yet started an AS course: 15 male, 12 female) were additionally involved in the pilot
study). Table 2 also shows the distribution of the students at the next two main stages of the study
(Data point 1 and Data point 2)
Table 2: Distribution (frequencies) of students according to gender and course
Gender
Maths Course
AS Trad
AS UoM
Total
PILOT
Male
Female
Total Pilot
144
70
214
55
44
99
199
114
313
Male
Female
Total DP1
769
511
1280
341
153
494
1110
664
1774
Male
Female
Total DP2
413
288
701
235
108
343
648
396
1044
DP1
DP2
‘Disposition to study mathematically demanding HE courses’
We begin with the results for the ‘disposition to study mathematically demanding subjects in HE’
scale, as this proved least problematic form a measurement point of view. Table 3 shows the items
and the pilot frequencies of responses: these items were informed by our pilot interviews with
students, and were all typical of the kinds of things students said to us in informal conversations and
interviews about their intentions and dispositions.
Table3: Items in the ‘Disposition to study mathematically demanding courses in HE’ instrument,
with pilot frequencies
4
PC coding
Item and Responses
ITEM NAME
Frequency (pilot)
4
3
2
1
0
9
Are you planning to study any more mathematics
courses or units after this AS course? [B1]
Yes
No
Don’t know
My preferred options for a course at university will
include: [B8]
A lot of mathematics
Quite a lot of mathematics
A moderate amount of mathematics
As little mathematics as possible
No mathematics
Don’t know
2
1
0
9
The amount of mathematics in my preferred options
for the course at university was: [B9]
Very important
Quite important
Not at all important
Don’t know
4
3
2
1
0
9
If I find out that my future course involves studying
more mathematics than I thought, this would make
me feel: [B10]
Very happy
Fairly happy
Indifferent
Unhappy
Very unhappy
Don’t know
2
0
1
0
2
1
If in the future I am studying a course involving
mathematics, then I would prefer it to be: [B11]
Familiar mathematics that I have already done
New mathematics that I have not learnt before
A mix of familiar and new mathematics
Don’t know
PLAN
197
82
69
AMOUNT
38
72
100
40
21
53
32 Missing
IMPORTANCE
53
131
76
62
Missing 32
FEELINGS
30
92
77
13
70
44
30 missing
MATH TYPE
98
22
184
62
Analysis and Results
Analysis was conducted on the full data sets for Data point 1 and 2 using a Partial Credit Model, in
the software ‘QUEST’. The fit statistics are better than acceptable for a case such as this, as shown
below in Table 4a,b for the pilot data and the main data points 1 and 2:
Table 4a: Measures (Thresholds) and fit statistics for the items of the MHE-disposition Scale at
Pilot
5
Item name
(number)
Plan (1)
Amount (7)
Import (8)
Feelings (9)
MathType(10)
Mean
SD
1
-1.25 (.25)
-2.44 (.38)
-0.94 (.31)
-3.00 (.41)
-0.88 (.28)
-0.01
0.69
Thresholds (SE)
2
3
-0.27 (.26)
-1.21 (.29)
0.41 (.24)
1.61 (.32)
-0.77 (.26)
0.26 (.26)
2.90 (.40)
4
1.95 (.31)
2.32 (.34)
Infit
MNSQ
1.06
0.78
0.88
1.17
1.08
0.99
0.16
Outfit
MNSQ
1.13
0.78
0.88
1.17
1.08
1.01
0.17
Table 4b: Fit statistics at DP1 and DP2
Item
Plan
Amount
Importance
Feelings
Math Type
Mean
SD
Infit MSQR
DP1
DP2
0.95
0.93
0.80
0.81
0.91
0.99
0.95
0.91
1.31
1.27
0.98
0.98
0.19
0.17
Outfit MSQR
DP1
0.98
0.79
0.90
0.96
1.37
1.00
0.22
DP2
0.97
0.82
1.02
0.94
1.34
1.02
0.19
The item map (Figure 1) for data point 1 shows how the students scores are separated by the items
in a very efficient way, any improvement in the measure would be at the cost of adding extra items
or refinements of codings, in order to reduce errors of estimation, probably n the middle of the
spread.
Figure 1 : Item map for HE maths instrument at data point one
--------------------------------------------------------------------------------------------------Each X = 12 students
Item Estimates (Thresholds) (number of item given in table 4
-------------------------------------------------------------------------------------------------|
XX
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3.0
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B10 .2 (Math type)
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XXXX
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B7 .4 (Amount) B9 .4 (feelings)
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2.0
XXXXXXXX
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X
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XXXXXXXXXXXXX
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X
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X
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B8 .2 (importance) B9 .3
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1.0
XXXXXXXXXXXXXXXX
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B7 .3
X
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XX
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XXX
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B1 .2
0.0
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6
XXXXXX
XXXXXXXXXXXXXXXXXXX
X
XX
XXXXXXXXXXXXX
XXXX
XX
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B10 .1
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-1.0
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XXXXXXX
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B1 .1
XX
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XXX
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B7 .2 B8 .1
X
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B9 .2
-2.0
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XXX
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B7 .1
X
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B9 .1
-3.0
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X
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---------------------------------------------------------------------------------------------------
The HE disposition measure
However, the story with the other measure was more problematic. The HE disposition scale
consisted of the following 4 items/response codes: see table 5
“family”
“self”
Table 5: Information for the items of the scale ‘Disposition to enter HE’
Item
Item name Question and
PC
Freq
Freq
description
Responses
Scorin PILOT DP1
g
My
Are you planning to go to
expectation
University?
Yes
2
261
1207
Depends
1
78
440
No
0
14
64
Family
(My family expects that )
Expectation
That I will go to university 2
272
1212
“friends”
Friends
Expectation
Different/conflicting
expectations
Indifferent ????
That I will not go
(My friends expect that )
That I will go to university
Different/conflicting
expectations
NO expectation
Don’t know
That I will not go
1
1
0
16
37
9
470
31
2
271
1013
1
1
1
0
28
23
655
10
44
7
Teachers’
expectation
(My Teachers expect that )
That I will go to university 2
“teachers”
Different/conflicting
expectations
NO expectation
Don’t know
That I will not go
1
1
1
0
264
852
24
13
844
11
10
Analysis here involved the use of the Partial Credit Model, for outcomes recorded in more than two
ordered response categories, i.e. by awarding ‘partial credit’ for responses that are neither correct
nor totally incorrect (Masters, 1999). In this case, partial credit was awarded in items with
‘intermediate’ levels of expectancy to go to university, e.g. the responses coded as ‘1’ in Table 2.
The results of the pilot analysis are shown below. As shown in Table 3 item fit analysis seems
adequate: all items infit statistics are within the acceptable limits expected in this kind of research.
However, the fit of the’ teachers’ item in DP1 is interesting: the relatively high Outfir suggests that
there are at least some students whose responses tot his item do not fit as well as the others. This
might be explained by the fact that the se students get less feedback form their teachers on this issue
than they do from friends and family, or that the feedback they get might be dissonant with their
own view of themselves. The current project does not have ambitions to further explore this
question, but the data are somewhat suggestive.
Table 6: Measures (Thresholds) and fit statistics for the items of the HE-disposition Scale at Pilot
and Data Point 1
Thresholds (SE)
Pilot
Item
Self
Family
Friends
Teachers
Mean
SD
1
-1.25
-1.56
-1.53
-1.25
0.00
0.19
(.50)
(.59)
(.56)
(.56)
2
1.75 (.37)
1.16 (.42)
1.34 (.39)
1.32 (.41)
DP1
-1.72
-2.69
-2.25
-4.13
0.00
0.32
1
(.25)
(.31)
(.31)
(.47)
2
2.08 (.15)
2.07 (.13)
2.96 (.15)
3.67 (.14)
Infit MSQR
Pilot
DP1
Outfit MSQR
Pilot
DP1
1.04
0.92
1.00
1.01
0.99
0.05
1.03
0.92
1.00
0.99
0.98
0.05
0.90
0.93
0.85
1.20
0.97
0.16
0.90
0.86
0.81
1.87
1.11
0.51
Figure 2 presents the actual maps of items difficulties and person’s ability in the common
constructed scale for the pilot data and DP1 analysis:
Figure 2a: The “HE Disposition” scale (pilot data) item map and fit
Each X represents
2 students
Item Estimates (Thresholds)
--------------------------------------------------------------------------------------------3.0
|
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XXXXXXXXXXXXXXXXXXXXXXX
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XX
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XX
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8
2.0
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Self.2
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X
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XXXXXXXXXX
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Frie.2 Teac.2
|
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Fam .2
1.0
XXXXXX
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X
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XXXXXXXXXXX
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X
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0.0
XXX
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XXXXXXX
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XXXX
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X
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-1.0
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Self.1 Teac.1
X
|
X
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Fam .1 Frie.1
|
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-2.0
|
XX
|
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X
|
---------------------------------------------------------------------------------------------------
An observation of the pilot results concerned the fact that the scale could not discriminate a lot for
the students of this sample since most of them appear to be gathered at the top end of the scale,
meaning that they have high disposition to enter HE. (It should also be noted that many students
appeared with perfect scores in this scale and their actual logit measure could therefore not be
calculated without making sample-assumptions). This was believed to be related to the timing of the
pilot at the end of AS year, and we expected to have more spread of responses at the beginning fo
the College course, which was the relevant moment for the main stage of the study. Therefore, the
decision was taken for this instrument to remain as it is and no changes to be made, apart from
some minor modification in the wording of the items to make them consistent.
The story of the measure during the next stage is shown in following figure, 2b.
Figure 2b: The “HE Disposition” scale (DP1 analysis) item map
Each X represents
20 students
Item Estimates (Thresholds)
--------------------------------------------------------------------------------------------------4.0
XXXXXXXXXXXXXXXXXXXX
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Teac.2
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3.0
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Frie.2
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XXXXXXXXXXXXXX
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9
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Self.2 Fam .2
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XXXXXXXXX
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1.0
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0.0
XXXXXXXXXXX
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-1.0
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XX
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Self.1
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-2.0
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Frie.1
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X
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Fam .1
|
-3.0
|
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-4.0
X
|
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Teac.1
--------------------------------------------------------------------------------------------------2.0
One notices in both these item maps some lack of separation and discrimination in the middle of the
scale, with many item thresholds grouped. This is suggestive of a ‘levelling’ in the sample, and one
might be able t establish some criteria for developing a hierarchy.
More important and problematic however, it seems that, contrary to our expectation, the measure
remained unable to discriminate the high end of the spectrum, which includes about 400 students in
our sample, who still get the highest score. It is too “easy” for our target groups. Hence the decision
was taken to try to extend the measure to include a little bit more difficulty by adding two more
difficult items at DP2. (See table 7).
10
Considering a new HE measure at DP2
Two new Likert type items were included in the instrument to improve the separability of the scale
These are in table 7, and the fit statistics (a surprise!) are in Table 8: the model seems to indicate a
bad fit to “Repeat” on both infit and outfit, suggesting that this misfit is not caused by a few offtarget responses but is probably an issue for students right across the measured spectrum.
The infit for the new item “Take it” is healthy: while the item has a slightly higher infit than the
others, it is also a hard item, and it is suggestive that this infit is related to the fact that a hard item
on a scale is vulnerable to a few off-target residuals. In this case the relatively high outfit for “Take
it” (1.45) is consistent with this interpretation. We therefore conclude that the measure benefits
form “take it” while “Repeat” potentially threatens the construct as a whole.
Table 7: two additional items for the HE disposition scale, “Repeat” and “take it”
:
Disagree
strongly
New1: Repeat “ I am prepared to repeat a year at college in
order to get into university, if necessary.”
1
New2: Take it “ If I was offered the career I wanted without
having to go to university, I would consider taking it. “
4
Disagree
Agree
Agree
strongly
2
3
4
3
2
1
NB: this item is Reversed coded
Table 8a: the fit statistics for the HE instrument at data point 2
--------------------------------------------------------------------------------------------------Item Fit
all on all (N = 1820 L = 6 Probability Level=0.50
--------------------------------------------------------------------------------------------------INFIT
MNSQ
0.56
0.63
0.71
0.83
1.00
1.20
1.40
1.60
1.80
-------------+---------+---------+---------+---------+---------+---------+---------+---------+----1 Self
*
.
|
.
2 repeat
.
|
.
*
3 take it
.
|
*
.
4 Fam
*.
|
.
5 Frie
.*
|
.
6 Teac
.
*
.
===================================================================================================
Table 8b: Fit statistics for analyses on different data sets for the HE disposition with and without
the 2 new items
Item
Self
Repeat
Take it
Family
Friends
Teachers
Mean
SD
DP1
0.90
0.93
0.85
1.20
0.97
0.16
Infit MSQR
DP2-A
DP2-B
0.86
0.62
1.56
1.10
0.88
0.76
0.89
0.78
0.99
1.22
0.96
0.97
0.17
0.34
DP2-C
0.75
1.19
0.83
0.86
1.13
0.95
0.19
DP1
0.90
0.86
0.81
1.87
1.11
0.51
Outfit MSQR
DP2-A
DP2-B
0.88
0.58
1.62
1.16
0.83
0.66
0.81
0.68
1.48
0.97
1.00
0.94
0.32
0.40
DP2-C
0.84
1.45
1.07
1.01
1.47
1.17
0.28
11
The decision to exclude the item from the measurement scale does have the effect of reducing the
difficulty of the scale as a whole however, and leads in the end to us having to deal with numbers of
cases of students with ‘full score’ still. Nevertheless the final scale including the new item has
better separability at the top end of the scale than the original scale, as is indicated in Figure 3:
notice that the responses ‘3’ and ‘4’ on this item are particularly critical!
Figure 3: Item map for HE disp - Analysis of Data Point 2- with the new item “Take it”
--------------------------------------------------------------------------------------------------Item Estimates (Thresholds)
Each X represents
12 students
7.0
|
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new2.4
6.0
|
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5.0
XXXXXXXXXXXX
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4.0
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new2.3
XXXXXXXXXXXXXXXXXXXX
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3.0
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XXXXXXXXXXXXXXX
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2.0
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Teachers.2
XXXXXXXXXXX
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Friends.2
1.0
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Self.2 Family.2
XXXXXXXX
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new2.2
|
0.0
XXXXXX
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-1.0
XXXXX
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Self.1
-2.0
XXX
|
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-3.0
X
|
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Family.1 Friends.1
|
|
Teachers.1
-4.0
X
|
---------------------------------------------------------------------------------------------------
The item “Repeat” raises another issue however, why does this item misfit so badly? We assume
that it taps into some other dimension, that there is a significant sample sub-group perhaps that is
responding to the item in ways that are distinct because of their particular situations. We speculate
that:
•
•
there may be some students for whom staying another year at College, for specific reasons
of their teachers, peer group, or their grant status, might find staying on unusually
acceptable or unacceptable;
there may be some relation between personal self-esteem and staying on that interferes
particularly with certain social groups.
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This is an empirical question and worthy of further research involving examination of the item
parameters for this item in various subgroups: we are testing for different background variable
effects and may be able to report this soon. First analyses suggest that Gender, Programme (use of
maths versus Traditional Mathematics) and class proxy variables are implicated in differential item
functioning.
Conclusions
We have constructed ‘robust’ measures of students’ dispositions (i) of their disposition to study
further mathematically-demanding subjects in HE, and (ii) of their commitment to study in HE,
based on short multiple choice items given to students studying AS level mathematics or Uses of
mathematics.
The first of these scales provides good separation of the sample, while the second proves
problematic at the top end: this led us to extend the scale with one harder item that fits tolerably,
and reduces the tendency to ‘total scores’ on the scale.
An interesting item that did not fit was also analysed and rejected as a possible addition: this leads
us to some further investigation of the construct, especially as concerns class, gender and other
aspects of background.
Discussion
We speculate now as to the possible use of this scale for other researchers’ purposes. The scalability
of an instrument like this is of course an empirical question: will the scale work for other sample sin
other situations? It may be that our group of students, predominantly following AS level courses
(though with some significant numbers of BTEC) have relative high commitments to HE entry, and
that the sale may actually be more suited to the population as a whole than to our own sample.
However, our own sample – chosen to reflect a group of students whose hold on education is
relatively tenuous - is relatively skewed to lower social classes in urban areas, and so this might
also be interpreted appropriately.
References
Bond T.G. and Fox, C.M. (2001). Applying the Rasch model: fundamental measurement in the
human sciences. Lawrence Erlbaum assoc: Mahwah, NJ.
Eley, M. G., & Meyer, J.H.F. (2004). Modelling the Influences on Learning Outcomes of Study
processes in University Mathematics, Higher Education, 47: 437-454.
Hoyles, C., Newman, K. and Noss, R. (2001). Changing patterns of transition from school to
university mathematics, International Journal of Mathematical Education in Science and
Technology, 32(6): 829-845.
Smith, A. (2004). Making Mathematics Count. HM Stationery Office, London.
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